Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO

Eman F. Mohommed; jalal Abd Jassim

doi:10.12688/f1000research.172910.2

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Research Article

Revised

Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO

[version 2; peer review: 1 approved, 1 not approved]

Eman F. Mohommed ¹, jalal Abd Jassim¹

PUBLISHED 15 May 2026

Author details Author details

¹ mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Baghdad Governorate, Iraq

Eman F. Mohommed
Roles: Methodology, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

jalal Abd Jassim
Roles: Funding Acquisition, Software, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

This paper studies a class of polyominoes. The new class is defined through a representation on the James abacus through nested chain which denoted Ω -nested Abacus. Depended on partition, beta number, nested chain we defined the structural condition this class. A local transformations, called SSPT-transformation and MSPT-transformation are formulated on the even chains. Then, we drive explicit formulas for the number of position in each chain, number of even chain and the total number of all chains. The structure of new class lead to enumeration formulas for the objects produced after application new transformation. To enhance further, based on these classes, generating functions are also being formulated by employing enumeration of combinatorial objects (ECO). In ECO method, each object is obtained from smaller object by making some local expansions. These local expansions are described in a simple way by a succession rule which can be translated into a function equation for the generating function.

Keywords

Polyominoes, Class, Enumeration, Succession-rule, Abacus diagram

Corresponding author: Eman F. Mohommed

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 Mohommed EF and Abd Jassim j. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Mohommed EF and Abd Jassim j. Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 2; peer review: 1 approved, 1 not approved]. F1000Research 2026, 15:11 (https://doi.org/10.12688/f1000research.172910.2) First published: 06 Jan 2026, 15:11 (https://doi.org/10.12688/f1000research.172910.1) Latest published: 15 May 2026, 15:11 (https://doi.org/10.12688/f1000research.172910.2)

Revised Amendments from Version 1

In this revised version, we have substantially improved the manuscript in response to the reviewers’ comments. Several definitions have been clarified, and all variables are now consistently defined and used throughout the paper.

The statements and proofs of Lemma 11 and Theorem 12 have been revised to explicitly include all necessary assumptions and to ensure that the arguments are self-contained. In addition, Corollary 13 and Theorem 14 have been rewritten with unified notation and clearer combinatorial reasoning.

The application of the ECO method has been reconstructed in greater detail, including a clearer description of the generating tree and the associated succession rule. Furthermore, the derivation of the generating function has been completely rewritten from first principles, with a more detailed and rigorous presentation.

Finally, the overall language, structure, and readability of the manuscript have been carefully improved to enhance clarity and coherence. We believe that these revisions significantly strengthen the mathematical rigor and presentation of the work.

See the authors' detailed response to the review by Hayder R. Hashim

Introduction

Polyominoes are finite configurations composed of unit squares, known as ominoes, that are joined edge to edge to form a connected interior, as illustrated in Figure 1.

Figure 1. 22-polyominoes (polyominoes with 22 connected ominoes).

An $n$ -polyomino (a polyomino consisting of $n$ ominoes) is defined up to translation, and the concept is commonly attributed to Golomb.¹ In contemporary research, n-polyominoes have attracted significant attention from computer scientists, physicists, mathematicians, and biologists. Despite extensive studies, enumerating n-polyominoes remains a challenging and unresolved problem in combinatorial geometry, representing one of its most fundamental open questions.^2–5 There is no closed-form equation for n-polyominoes, and the problem has been solved up to n < 56.³ No closed-form expression for the enumeration of $n$ -polyominoes is currently known. Due to the complexity of this problem, several simpler subclasses of $n$ -polyominoes have been formulated and extensively examined in the literature.^6–8 E.F. provide a new representation of n-polyominoes (Plyominoes with n ominoe) using one of the graphical representations of partition (James-diagram) called nested chain abacus (N.C.A.) with $b$ -columns and $d$ -rows.⁹ The Nested Chain Abacus (N.C.A.) offers a novel framework for representing any $n$ connected omino, including empty ones (holes), through the use of a beta number. In this new representation, $n$ -polyominoes are organised into a series of nested chains.¹⁰ A Nested Chain Abacus (N.C.A.) is an $n$ -polyomino inscribed within a James diagram, consisting of both outer and inner chains. The inner chains are numbered from 1 to $n$ , where $n$ is a positive integer, and chain 1 is the innermost chain. No intersections occur among any of the chains. Through this new representation, and for the first time, each n-polyomino has been systematically associated with a unique code. Next Figure 2 illustrates an example of an N.C.A. with six columns, five rows, and three chains.

Figure 2. Nested chain abacus with 6 columns, 5 rows and 3 chains.

Terminologies and definition

This section introduced some importuned definitions:

Definition 1.

⁴A partition of a positive integer, $k$ , is a sequence of integers $λ_{1}, λ_{2}, \dots, λ_{n}$ such that $λ_{1} \geq λ_{2} \geq \dots \geq λ_{n}$ and $\sum_{i = 1}^{n} λ_{i} = k$ .

Example 1.

$λ = (6, 6, 3, 2, 1, 1)$ is a partition of 19

Definition 2.

⁴A sequence of positive numbers { $β_{1}, β_{2}, \dots, β_{k}}$ called beta number such that $β_{i} = λ_{i} + k - i$ and $1 \leq i \leq k$ .

Example 2.

⁴Consider of Example 1, $λ = (6, 6, 3, 2, 1, 1)$ then beta number of $λ$ is

β_{i} = λ_{i} + b - i, where i = 1, 2, 3, 4, 5, 6

And hence

β_{1} = 11, β_{2} = 10, β_{3} = 6, β_{4} = 4, β_{5} = 2, β_{6} = 1

Thus beta number sequence is

{11, 10, 6, 4, 2, 1} .

Definition 3.

⁴James Abacus is a graphical representation of a partition of any integer number using beta numbers.

Consider Example 1, the James Abacus with 1,2,3,4,6,7,10,11 beta position as show in next Figure 3.

Definintion 4.

⁹A new representation of n-polyominoes (Plyominoes with n ominoe) using one of the graphical representations of partition (James-diagram) called nested chain abacus (N.C.A.) with b-columns and d-rows.

Next Figure 4 gives an example of N.C.A. with 3 chains represented of 94, where the beta number sequence is {0,5,6,7,8,9,10,11,14,15,16,19,20,21,24}.

Figure 3. James’s Abacus with 3 columns, 4 rows and 6 beta.

Figure 4. N.C.A. with 5 columns ( $b = 5$ ) and 3 chains represented of 94, where the beta number sequence is ${$ 0,5,6,7,8,9,10,11,14,15,16,19,20,21,24}.

Definition 5:

A connected chain is a chain with only beta positions.

Next Figure 5 gives an example of N.C.A. represented by a beta sequence {0,1,2,3,4,5,7,8,9,10,12,14,15,19,20,21,22,23,24} with one connected chain (chain 1).

Figure 5. N.C.A. with 1 connected chains (chain 3).

Definition 6.

Let $β_{i}, β_{j}$ be two beta numbers in N.C.A. then $β_{i}, β_{j}$ are connected iff

1- $| β_{i} - β_{j} | = 1$ if $β_{i}, β_{j}$ Located in the same row.
2- $| β_{i} - β_{j} | = b if β_{i}, β_{j} Located in the same column$ .

For example, the beta set

{0, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24} .

From Figure 4 is a connected.

Definition 7.

Each nested chain abacus (N.C.A.) with $b$ columns and b rows is called a polyamines if every two beta numbers are connected.

Definition 8.

Let N.C.A. be a nested chain abacus consist of $b$ column and $d$ rows, and let $C_{1}, C_{2}, \dots, C_{r}$ be its chains, order from the innermost chain to the outermost chain. The NCA is called an $Ω$ -nested Abacus if the following condition are satisfied

1- First chain (initial chain) $C_{1}$ which consist exactly one position.
- a. If $C_{1}$ consist exactly one beta position, then $C_{2}$ has at least five beta positions.
- b. If the position in $C_{1}$ is empty position, then $C_{2}$ consist $k$ beta positions, where $1 \leq k \leq 8$
2- Every even chain ( $C_{2 t})$ may contain both beta position and empty beta position.
3- Every odd chain ( $C_{2 t - 1})$ forms a connected chain.

Any polyomino represented by such a configuration is called an $Ω$ -nested Abacus polyomino.

Figure 6 gives an example of $Ω$ -nested Abacus with four chains, where

Not Every polyamines inscribed in a nested chain-abacus is $Ω$ -nested Abacus.

Figure 4 given an example of N.C.A. but not $Ω$ -nested Abacus, while Figure 6 given an example of $Ω$ -nested Abacus.

Not. Throughout our result the underlying new family ( $Ω$ -nested Abacus) assumed $b = d$ where $b$ is odd number. Since every odd chain is completely filled, so only even chain well be contribute to the generating process.

The algorithm construct to generate classes of new family depended on the following transformation.

Figure 6. $Ω$ -nested Abacus with four chains.

Definition 9.

Single – beta transformation (SPT)

Let an $Ω$ -nested Abacus with $b$ column, $d$ rows and $C_{i}$ chains. Suppose that a beta number $β = (m - 1) b + (j - 1)$ in chain $C_{i}$ is located in row $m$ and column $j$ such that $1 \leq m \leq d$ , $1 \leq j \leq b$ and $1 \leq i \leq r$ . A single beta transformation (SPT) in chain $C_{i}$ is local transformation ( $β \to \overset{`}{β}$ ) within the same chain where

\overset{`}{β} = {\begin{matrix} (m - 1) b + (j - 2) & if & i + 1 \leq m \leq (r - i + 1), j = b - i + 1 \\ mb + (j - 1) & if & i \leq m \leq (r - i), j = i \\ (m - 1) b + j & if & m = i, i + 1 < j \leq b - i + 1 \\ (m - 2) b + (j - 1) & if & m = d - i + 1, i \leq j < b - i . \end{matrix}

Remark 10.

The maximal number of transformations in chain $n$ is equal to the number of positions in the chain, where $n > 1$ .

Figure 7 illustrates Single – beta transformation (SPT)

Not.

1- SSpt is a SPT transformation in one chain.
2- MSpt is a SPT transformation in all chains.

Figure 7. Illustrates Single – beta transformation (SPT) on $Ω$ -nested Abacus.

Enumeration of the $Ω$ -nested Abacus class

Next, the enumeration of Ω-nested Abacus with b columns and d rows is presented.

Lemma 11.

Let an $Ω$ -nested Abacus with $b$ column, $d$ rows and $C_{r}$ chains then, the number of position in any chains is $| C_{i} | = 8 (n - 1) .$

Proof.

Based on Definition 9 $C_{1}$ consists of a single position. Then, the successive chain form a discrete ring around the single position with four side increments and four corner position. Then $| C_{2} | = 8$ , thus $| C_{n} | = | C_{n - 1} | + 8$ where $1 \leq n \leq r$ . Thus

| C_{n} | = 8 (n - 1)

Where $n > 1$ .

Corollary 12.

Based on Remark 10 and Lemma 11, the number of (SSPT) transformation in chain $C_{n}$ is equal to $8 (n - 1)$ .

Lemma 13.

Let Ω-nested Abacus be a polyomino intercept in N.C.A. with $b$ columns and $d$ rows such that $b = d$ and $b$ is odd. Then the total number of chain is $\frac{b + 1}{2}$ .

Proof.

Based on Ω-nested Abacus structure is symmetric with respect to a central column (central chain $C_{n}$ ) then for all i-th chains, the boundary columns must satisfy $i \leq b - i + 1$ , thus

$2 i \leq b - 1 \to i \leq \frac{b + 1}{2}$ . Then every Ω-nested Abacus with $\frac{b + 1}{2}$ chains.

Proposition 14.

Let Ω-nested Abacus be a polyomino intercept in NCA with $b$ columns and $d$ rows such that $b = d$ and $b$ is odd. Then the number of even chain is $\frac{b - 1}{4}$ .

Provided that $b \equiv 1 (mod 4) .$

Proof.

Based on Lemma 13 the number of chain in Ω-nested Abacus is $t = \frac{b + 1}{2}$ . The even chains among $1, 2, \dots, t$ are $2, 4, \dots, ⌊ \frac{t}{2} ⌋$ . Since $t = \frac{b + 1}{2}$ is even then $\frac{t}{2} = \frac{\frac{b + 1}{2}}{2}$ is even since the first chain is odd and fixed. Then there are Since $t = \frac{b + 1}{2}$ is even then $\frac{t}{2} = \frac{\frac{b + 1}{2} - 1}{2} = \frac{b - 1}{4}$ , then the number of even chain is $\frac{b - 1}{4}$ .

Next we found the number of Ω-nested Abacus if we application SSPT-Transformation in chain $i$ .

Lemma 15:

Let Ω-nested Abacus be a polyomino intercept in NCA with $b$ columns and $d$ rows such that $b = d$ , $b$ is odd and let $n$ be even number. Then the number of Ω-nested Abacus generating by employ SSPT-Transformation exactly chain $n$ is $8 (n - 1) .$

Proof.

By Lemma 11 the number of admissible positions in chains $n$ is $8 (n - 1)$ . Since move each beta position yields a distinct Ω-nested Abacus under SSPT-transformation obtain in chain n with $8 (n - 1)$ position is $8 (n - 1)$ .

Theorem 16:

Let Ω-nested Abacus be a polyomino intercept in N.C.A. with $b$ columns and $d$ rows such that $b = d$ , $b$ is odd and let $n$ be even number. Then the number of Ω-nested Abacus generating by employ SSPT-Transformation is

\sum_{\forall n} (\binom{R}{(n - 1) 8})

Where $R = \sum_{n}^{⌊ \frac{b + 1}{4} ⌋} (n - 1) 8$ , $b \equiv 1 (mod 4)$

Proof.

Based on Lemma 10, the number of positions in chain n is $(n - 1) 2^{3}$ by Lemma 11 and Definition 8, there are $\frac{b - 1}{4}$ chain with beta and empty beta position in Ω-nested Abacus. The maximal number of transformations in chain $n$ is equal to $((n - 1) 8)$ . Since each move generates a new Abacus (polyominoes), thus there are $((n - 1) 8)$ of Ω-nested Abacus generated by employing SSPt transformation. As a result of this, there are

\sum_{\forall n} (\binom{R}{(n - 1) 8})

Ω-nested Abacus.

Theorem 17.

\sum_{n = 2}^{⌊ \frac{b + 1}{4} ⌋} 8 (n - 1) .

Where $b \equiv 1 (mod 4)$

Proof.

Based on Lemma 11 the number of positions in chain $n$ is 8(n−1), any position in the chain will be determines one distinct SSPT in that chain. Thus all even chain $C_{2}, C_{4}, \dots,$ gives $\sum_{n = 2}^{⌊ \frac{b + 1}{4} ⌋} 8 (n - 1)$ .

Theorem 18.

\prod_{\forall n} 8 (n - 1)

Where $b \equiv 1 (mod 4)$ and $n = 2, 4, \dots . ., \frac{b = 1}{4}$

Proof.

Based on Lemma 11, the number of admissible beta and empty eta position in chain n is $8 (n - 1)$ . Since the odd chain are completely filled, the remain fixed and do not contribute to the transformation. By Lemma 15 the number of Ω-nested Abacus generating by employ SSPT-Transformation exactly chain $n$ is $8 (n - 1) .$ Since the MSPT-transformation acts simultaneous on all even chain. A generated Ω-nested Abacus is obtain by selecting one admissible position in each chain. The choice in one even chain does not restrict the admissible choices in any other even chain. Then the number of Ω-nested Abacus can be generated after application SSPT-Transformation is $\prod \forall n 8 (n - 1)$ .

Generating function with respect to chains

In this section, the method described in^11,12 is employed to enumerate the Ω-nested Abacus class, representing polyominoes inscribed within a James diagram. The ECO (Enumerating Combinatorial Objects) method has previously been applied to the enumeration of various polyomino classes.¹² This approach is based on a succession rule.

Generating Function (G.F.)

Let $C_{2 k}$ be even chain of Ω-nested Abacus, based on Lemma 11 the number of positions in any chain is

M_{k} = ⌈ C_{2 k} ⌉ = 8 (n - 1) .

We application SSPT-Transformation inside this chains such that no previously select position may be chose again. Thus, in initial stage there are $M_{k}$ choices ( $M_{k}$ of Ω-nested Abacus), after one insertion we have only $M_{k} - 1$ position ( $M_{k} - 1$ of Ω-nested Abacus), after two distinct insertion there are only $M_{k} - 2$ position ( $M_{k} - 2$ of Ω-nested Abacus as shown in Figure 8), and so on. Thus, there are

a_{n}^{k} = M_{k} (M_{k} - 1) (M_{k} - 2) \dots (M_{k} - n + 1), 2 \leq n \leq M_{k}

Figure 8. First and second level of $ϑ$ using $Ω$ -nested Abacus class with 5 columns, 5 rows and 3 chain.

Above process is encoded by a generating tree. The root labeled by $M_{k}$ , each node will be produces $j$ children, every one of $j$ children will be product $j - 1$ . Thus the local succession rule is

ϑ = {\begin{matrix} M_{k} \\ j & \to & {(j - 1)}^{j} . 1 \leq j \leq M_{k} \end{matrix}

Assume that $a_{n + 1}^{k}$ denoted the number of nodes at level $n$ $ϑ$ yield the recurrence

a_{n + 1}^{k} = (M_{k} - n) a_{n}^{k}

Where $a_{0}^{k} = 1, 2 \leq n \leq M_{k}$ .

a_{n}^{k} = \frac{M_{k}!}{(M_{k} - n)!} .

Using the ordinary level-generating polynomial of the even chain

C_{k} (x) = \sum_{n = 2}^{M_{k}} (M_{k})_{n} x^{n}

C_{k} (x) = \sum_{n = 2}^{M_{k}} \frac{{(M_{k})}_{n}}{n!} x^{n}

Since

\frac{{(M_{k})}_{n}}{n!} = \frac{M_{k}!}{(M_{k} - n)! n!} = (\begin{matrix} M_{k} \\ n \end{matrix})

then,

C_{k} (x) = \sum_{n = 2}^{M_{k}} (\begin{matrix} M_{k} \\ n \end{matrix}) x^{n} = (1 + x)^{M_{k}}

C_{k} (x) = \sum_{n = 2}^{M_{k}} (\begin{matrix} M_{k} \\ n \end{matrix}) x^{n} = (1 + x)^{8 (2 k - 1)}

Then the generating function

C_{k} (z, x) = \sum_{k = 2}^{m} z^{k} (C_{k} (x)) = \sum_{n = 2}^{M_{k}} z^{k} (1 + x)^{8 (2 k - 1)}

Where $m = ⌊ \frac{b + 1}{4} ⌋$

Thus

C_{k} (z, x) = \sum_{n = 2}^{M_{k}} (z (1 + x)^{16})^{k}

C_{k} (z, x) = z (1 + x)^{8} \frac{1 - (z (1 + x)^{16})^{m}}{1 - z (1 + x)^{16}}

$C_{k} (z, x)$ is generating function to enumerated the number of Ω-nested Abacus.

Conclusion

This study examines a class of polyominoes known as the Ω-nested Abacus. Initially, we provide a characterization of a new class based on specific geometric constraints defined by rows, columns, and chains. A series of operations on polyominoes is then introduced using a partition-theoretic construct known as the beta number. Furthermore, a set of operations on polyominoes is developed using a partition-theoretic construct known as the beta number, allowing for localized transformation of the structure. Furthermore, A succession rule is formulated based on generating tree techniques to describe the growth of these object. Based on this framework a recursive method is established for the systematic generation of Ω-nested Abacus configuration of a given size through the use of generating trees.

Ethical approval

Ethical approval was not required for this study.

Data availability statement

No data availability with this atrial as the study is purely theoretical and does not involve the generating analysis or use of any datasets.

References

1. Golomb S: Checkerboards and polyominoes. Am. Math. Mon. 1954; 61(10): 675682. Publisher Full Text
2. Flórez R, José Ramírez L, Villamizar D: Restricted bargraphs and unimodal compositions. Journal of Combinatorial Theory, Series A. 2024; 208: 105934. Publisher Full Text
3. Arabi A, Belbachir H, Dubernard JP: Enumeration of plateau polycubes with respect to their lateral area. Indian Journal of Pure and Applied Mathematics. 2024; 55: 538–554. Publisher Full Text
4. Jensen C: Counting polyominoes: a parallel implementation for cluster Computing. Proceedings of the 2003 International Conference on Computational Science: Part III, ICCS’03. Springer-Verlag; 2003; pp. 203–212.
5. Castiglione G, Restivo A: Ordering and convex polyominoes. the International Conference on Machines, Computations, and Universality. Berlin, Heidelberg: Springer Berlin Heidelberg; September 2004; pp. 128–139.
6. Del Lungo A: Polyominoes defined by two vectors. Theor. Comput. Sci. 1994; 127(1): 187–198. Publisher Full Text
7. Castiglione G, Frosini A, Restivo A, et al.: A tomographical characterisation of convex polyominoes. Poitiers, Proc. of Discrete Geometry for Computer Imagery 12th International Conference, DGCI 2005, in: Lecture Notes in Computer. 2005; pp. 115–125.
8. Castiglione G, Restivo A: Reconstruction of L-convex polyominoes. Electron. Notes Discret. Math. 2003; 12: 290–301. Publisher Full Text
9. Mohommed EF: Topological Structure of Nested Chain Abacus. Iraqi Journal of Science. 2020; 153–160. Publisher Full Text
10. Mohommed EF: Constructing a Nested Chain in James Abacus Diagram. J. Phys. Conf. Ser. 2019; 1294(3): 032019. IOP Publishing. Publisher Full Text
11. grammars to ECO systems. Theor. Comput. Sci. 2004; 57–95.
12. Banderier C, Wallner M: Lattice paths with catastrophes. Discrete Mathematics & Theoretical Computer Science. 2017; 19(1). Analysis of Algorithms. Publisher Full Text

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Version 2

VERSION 2 PUBLISHED 06 Jan 2026

Author details Author details

¹ mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Baghdad Governorate, Iraq

Eman F. Mohommed
Roles: Methodology, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

jalal Abd Jassim
Roles: Funding Acquisition, Software, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

Article Versions (2)

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Published: 15 May 2026, 15:11

https://doi.org/10.12688/f1000research.172910.2

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Published: 06 Jan 2026, 15:11

https://doi.org/10.12688/f1000research.172910.1

© 2026 Mohommed EF and Abd Jassim j. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mohommed EF and Abd Jassim j. Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 2; peer review: 1 approved, 1 not approved]. F1000Research 2026, 15:11 (https://doi.org/10.12688/f1000research.172910.2)

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Reviewer Report 25 May 2026

Hayder R. Hashim, University of Kufa, Kufa, Iraq

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https://doi.org/10.5256/f1000research.197917.r485075

The authors have revised the manuscript in accordance with my comments. My only remaining suggestion ... Continue reading

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Reviewer Report 05 Mar 2026

Hayder R. Hashim, University of Kufa, Kufa, Iraq

Approved with Reservations

https://doi.org/10.5256/f1000research.190675.r462385

The manuscript proposes a new class of polyominoes defined by a graphical representation referred to as the James Abacus or Nested Chain Abacus and attempts to enumerate this class using combinatorial transformations and the ECO method.

Since the results in the paper are interesting in number theory, I suggest accepting the paper after revising it according to the following remarks as the paper would benefit from improvements in clarity, organization, and presentation.

1. Lemma 11 and Theorem 12 refer to external theorems without clearly stating assumptions or applicability.
2. Corollary 13 and Theorem 14 contain inconsistent notation such as mixing b,p,r,e,d and unclear counting logic.
3. Polish the language carefully.
4. Every lemma and theorem must include a complete, self-contained proof.
5. All variables must be clearly defined and used consistently.
6. If results rely on previous publications, the exact statements must be cited and restated.
7. Try to reconstruct in more details the ECO method application properly.
8. Redo or rewrite the derivation of the generating function theorem from first principles, with a more clear form.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Number Theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Author Response 15 May 2026

Eman Mohommed, mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Iraq

15 May 2026

Author Response

We thank the reviewer for his comments
Comment 1:
Lemma 11 and theorem 12 refer to external theorems without clearly stating assumptions of applicability
Response: in the revised version, we ... Continue reading We thank the reviewer for his comments
Comment 1:
Lemma 11 and theorem 12 refer to external theorems without clearly stating assumptions of applicability
Response: in the revised version, we have explicitly stated all required assumptions and clarified the applicability of the referenced result. In addition, we have added several lemmas and theorems related with lemma 11 and theorem 12 to ensure that lemma 11 and theorem 12 are fully self-contained and mathematically precise.
Comment 2:
Corollary 13 and Theorem 14 contain inconsistent notation, such as mixing b, p, r, e, d, and unclear counting logic
Response: we appreciate this observation. The notation has been carefully revised and unified throughout the paper. All variables are now clearly defined and used consistently.
Comment 3:
Polish the language carefully
Response: all manuscripts have been proofread and language has been improved.
Comment 4:
Every lemma and theorem must include a complete, self-contained proof.
Response: In the new version, all lemmas and theorems now include complete proof. Additional explanations have been added where necessary.
Comment 5:
All variables must be clearly defined and used consistently.
Response:
All variables have been systematically reviewed and defined at their first occurrence and then has been standardized across the paper to ensure consistency.
Comment 6:
If results rely on previous publications, the exact statements must be cited and restated
Response: we revised all relevant sections to explicitly cite the original source.
Comment 7,8:
Try to reconstruct in more detail the proper application of the ECO method.
Redo or rewrite the derivation of the generating function theorem from first.
Response: following the reviewer's suggestion, we have completely rewritten the derivation of the generating function.
We thank the reviewer for his comments
Comment 1:
Lemma 11 and theorem 12 refer to external theorems without clearly stating assumptions of applicability
Response: in the revised version, we have explicitly stated all required assumptions and clarified the applicability of the referenced result. In addition, we have added several lemmas and theorems related with lemma 11 and theorem 12 to ensure that lemma 11 and theorem 12 are fully self-contained and mathematically precise.
Comment 2:
Corollary 13 and Theorem 14 contain inconsistent notation, such as mixing b, p, r, e, d, and unclear counting logic
Response: we appreciate this observation. The notation has been carefully revised and unified throughout the paper. All variables are now clearly defined and used consistently.
Comment 3:
Polish the language carefully
Response: all manuscripts have been proofread and language has been improved.
Comment 4:
Every lemma and theorem must include a complete, self-contained proof.
Response: In the new version, all lemmas and theorems now include complete proof. Additional explanations have been added where necessary.
Comment 5:
All variables must be clearly defined and used consistently.
Response:
All variables have been systematically reviewed and defined at their first occurrence and then has been standardized across the paper to ensure consistency.
Comment 6:
If results rely on previous publications, the exact statements must be cited and restated
Response: we revised all relevant sections to explicitly cite the original source.
Comment 7,8:
Try to reconstruct in more detail the proper application of the ECO method.
Redo or rewrite the derivation of the generating function theorem from first.
Response: following the reviewer's suggestion, we have completely rewritten the derivation of the generating function.
Competing Interests: No competing interests were disclosed. Close
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Author Response 15 May 2026

Eman Mohommed, mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Iraq

15 May 2026

Author Response

We thank the reviewer for his comments
Comment 1:
Lemma 11 and theorem 12 refer to external theorems without clearly stating assumptions of applicability
Response: in the revised version, we ... Continue reading We thank the reviewer for his comments
Comment 1:
Lemma 11 and theorem 12 refer to external theorems without clearly stating assumptions of applicability
Response: in the revised version, we have explicitly stated all required assumptions and clarified the applicability of the referenced result. In addition, we have added several lemmas and theorems related with lemma 11 and theorem 12 to ensure that lemma 11 and theorem 12 are fully self-contained and mathematically precise.
Comment 2:
Corollary 13 and Theorem 14 contain inconsistent notation, such as mixing b, p, r, e, d, and unclear counting logic
Response: we appreciate this observation. The notation has been carefully revised and unified throughout the paper. All variables are now clearly defined and used consistently.
Comment 3:
Polish the language carefully
Response: all manuscripts have been proofread and language has been improved.
Comment 4:
Every lemma and theorem must include a complete, self-contained proof.
Response: In the new version, all lemmas and theorems now include complete proof. Additional explanations have been added where necessary.
Comment 5:
All variables must be clearly defined and used consistently.
Response:
All variables have been systematically reviewed and defined at their first occurrence and then has been standardized across the paper to ensure consistency.
Comment 6:
If results rely on previous publications, the exact statements must be cited and restated
Response: we revised all relevant sections to explicitly cite the original source.
Comment 7,8:
Try to reconstruct in more detail the proper application of the ECO method.
Redo or rewrite the derivation of the generating function theorem from first.
Response: following the reviewer's suggestion, we have completely rewritten the derivation of the generating function.
We thank the reviewer for his comments
Comment 1:
Lemma 11 and theorem 12 refer to external theorems without clearly stating assumptions of applicability
Response: in the revised version, we have explicitly stated all required assumptions and clarified the applicability of the referenced result. In addition, we have added several lemmas and theorems related with lemma 11 and theorem 12 to ensure that lemma 11 and theorem 12 are fully self-contained and mathematically precise.
Comment 2:
Corollary 13 and Theorem 14 contain inconsistent notation, such as mixing b, p, r, e, d, and unclear counting logic
Response: we appreciate this observation. The notation has been carefully revised and unified throughout the paper. All variables are now clearly defined and used consistently.
Comment 3:
Polish the language carefully
Response: all manuscripts have been proofread and language has been improved.
Comment 4:
Every lemma and theorem must include a complete, self-contained proof.
Response: In the new version, all lemmas and theorems now include complete proof. Additional explanations have been added where necessary.
Comment 5:
All variables must be clearly defined and used consistently.
Response:
All variables have been systematically reviewed and defined at their first occurrence and then has been standardized across the paper to ensure consistency.
Comment 6:
If results rely on previous publications, the exact statements must be cited and restated
Response: we revised all relevant sections to explicitly cite the original source.
Comment 7,8:
Try to reconstruct in more detail the proper application of the ECO method.
Redo or rewrite the derivation of the generating function theorem from first.
Response: following the reviewer's suggestion, we have completely rewritten the derivation of the generating function.
Competing Interests: No competing interests were disclosed. Close
Report a concern

Views

Reviewer Report 16 Jan 2026

Ayesha Qureshi, Sabanci Universitesi (Ringgold ID: 52991), Istanbul, Istanbul, Turkey

Not Approved

https://doi.org/10.5256/f1000research.190675.r450050

This paper introduces a class of polyominoes defined via a nested chain abacus representation and aims to enumerate this class using transformations and the ECO method. While the topic may be of interest, the manuscript in its current form lacks sufficient mathematical clarity and rigor.

Several definitions are unclear or inconsistent, and many results are stated without complete or verifiable proofs. In particular, the use of the ECO method and the derivation of the generating function require a more precise and rigorous formulation. Additionally, the presentation and language need substantial improvement. A major revision addressing these issues would be necessary before the conclusions can be considered supported by the results.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Algebra

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

CITE

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Version 2

VERSION 2 PUBLISHED 06 Jan 2026

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Ayesha Qureshi, Sabanci Universitesi (Ringgold ID: 52991), Istanbul, Turkey
Hayder R. Hashim, University of Kufa, Kufa, Iraq

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1 Views

25 May 2026 | for Version 2

Hayder R. Hashim, University of Kufa, Kufa, Iraq

1 Views Cite this report Responses(0)

Approved

The authors have revised the manuscript in accordance with my comments. My only remaining suggestion to the authors is to carefully polish the language. Otherwise, I recommend acceptance for indexing.

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Number Theory and cryptography

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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12 Views

05 Mar 2026 | for Version 1

Hayder R. Hashim, University of Kufa, Kufa, Iraq

12 Views Cite this report Responses(1)

Approved With Reservations

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Number Theory

Respond to this report

Responses (1)

Author Response

15 May 2026

Eman Mohommed, mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Iraq

We thank the reviewer for his comments
Comment 1:
Lemma 11 and theorem 12 refer to external theorems without clearly stating assumptions of applicability
Response: in the revised version, we have explicitly stated all required assumptions and clarified the applicability of the referenced result. In addition, we have added several lemmas and theorems related with lemma 11 and theorem 12 to ensure that lemma 11 and theorem 12 are fully self-contained and mathematically precise.
Comment 2:
Corollary 13 and Theorem 14 contain inconsistent notation, such as mixing b, p, r, e, d, and unclear counting logic
Response: we appreciate this observation. The notation has been carefully revised and unified throughout the paper. All variables are now clearly defined and used consistently.
Comment 3:
Polish the language carefully
Response: all manuscripts have been proofread and language has been improved.
Comment 4:
Every lemma and theorem must include a complete, self-contained proof.
Response: In the new version, all lemmas and theorems now include complete proof. Additional explanations have been added where necessary.
Comment 5:
All variables must be clearly defined and used consistently.
Response:
All variables have been systematically reviewed and defined at their first occurrence and then has been standardized across the paper to ensure consistency.
Comment 6:
If results rely on previous publications, the exact statements must be cited and restated
Response: we revised all relevant sections to explicitly cite the original source.
Comment 7,8:
Try to reconstruct in more detail the proper application of the ECO method.
Redo or rewrite the derivation of the generating function theorem from first.
Response: following the reviewer's suggestion, we have completely rewritten the derivation of the generating function.

View more View less

Competing Interests

No competing interests were disclosed.

Back to all reports

Reviewer Report

31 Views

16 Jan 2026 | for Version 1

Ayesha Qureshi, Sabanci Universitesi (Ringgold ID: 52991), Istanbul, Istanbul, Turkey

31 Views Cite this report Responses(0)

Not Approved

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Algebra

Respond to this report

Responses (0)

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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[2] 2. Flórez R, José Ramírez L, Villamizar D: Restricted bargraphs and unimodal compositions. Journal of Combinatorial Theory, Series A. 2024; 208: 105934. Publisher Full Text

[3] 3. Arabi A, Belbachir H, Dubernard JP: Enumeration of plateau polycubes with respect to their lateral area. Indian Journal of Pure and Applied Mathematics. 2024; 55: 538–554. Publisher Full Text

[4] 4. Jensen C: Counting polyominoes: a parallel implementation for cluster Computing. Proceedings of the 2003 International Conference on Computational Science: Part III, ICCS’03. Springer-Verlag; 2003; pp. 203–212.

[5] 5. Castiglione G, Restivo A: Ordering and convex polyominoes. the International Conference on Machines, Computations, and Universality. Berlin, Heidelberg: Springer Berlin Heidelberg; September 2004; pp. 128–139.

[6] 6. Del Lungo A: Polyominoes defined by two vectors. Theor. Comput. Sci. 1994; 127(1): 187–198. Publisher Full Text

[7] 7. Castiglione G, Frosini A, Restivo A, et al.: A tomographical characterisation of convex polyominoes. Poitiers, Proc. of Discrete Geometry for Computer Imagery 12th International Conference, DGCI 2005, in: Lecture Notes in Computer. 2005; pp. 115–125.

[8] 8. Castiglione G, Restivo A: Reconstruction of L-convex polyominoes. Electron. Notes Discret. Math. 2003; 12: 290–301. Publisher Full Text

[9] 9. Mohommed EF: Topological Structure of Nested Chain Abacus. Iraqi Journal of Science. 2020; 153–160. Publisher Full Text

[10] 10. Mohommed EF: Constructing a Nested Chain in James Abacus Diagram. J. Phys. Conf. Ser. 2019; 1294(3): 032019. IOP Publishing. Publisher Full Text

[11] 11. grammars to ECO systems. Theor. Comput. Sci. 2004; 57–95.

[12] 12. Banderier C, Wallner M: Lattice paths with catastrophes. Discrete Mathematics & Theoretical Computer Science. 2017; 19(1). Analysis of Algorithms. Publisher Full Text

Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO

Abstract

Keywords

Revised Amendments from Version 1

Introduction

Figure 1. 22-polyominoes (polyominoes with 22 connected ominoes).

Figure 2. Nested chain abacus with 6 columns, 5 rows and 3 chains.

Terminologies and definition

Definition 1.

Example 1.

Definition 2.

Example 2.

Definition 3.

Definintion 4.

Figure 3. James’s Abacus with 3 columns, 4 rows and 6 beta.

Figure 4. N.C.A. with 5 columns ( b=5 ) and 3 chains represented of 94, where the beta number sequence is { 0,5,6,7,8,9,10,11,14,15,16,19,20,21,24}.

Definition 5:

Figure 5. N.C.A. with 1 connected chains (chain 3).

Definition 6.

Definition 7.

Definition 8.

Figure 6. Ω -nested Abacus with four chains.

Definition 9.

Remark 10.

Figure 7. Illustrates Single – beta transformation (SPT) on Ω -nested Abacus.

Enumeration of the Ω -nested Abacus class

Lemma 11.

Proof.

Corollary 12.

Lemma 13.

Proof.

Proposition 14.

Proof.

Lemma 15:

Proof.

Theorem 16:

Proof.

Theorem 17.

Proof.

Theorem 18.

Proof.

Generating function with respect to chains

Generating Function (G.F.)

Figure 8. First and second level of ϑ using Ω -nested Abacus class with 5 columns, 5 rows and 3 chain.

Conclusion

Ethical approval

Data availability statement

References

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Figure 4. N.C.A. with 5 columns ( $b = 5$ ) and 3 chains represented of 94, where the beta number sequence is ${$ 0,5,6,7,8,9,10,11,14,15,16,19,20,21,24}.

Figure 6. $Ω$ -nested Abacus with four chains.

Figure 7. Illustrates Single – beta transformation (SPT) on $Ω$ -nested Abacus.

Enumeration of the $Ω$ -nested Abacus class

Figure 8. First and second level of $ϑ$ using $Ω$ -nested Abacus class with 5 columns, 5 rows and 3 chain.